What is the purpose of "reshaping it into the shape the network expects and scaling it so that all values are in the [0, 1] interval."?
The purpose of rescaling  gradient descent problems is to reframe the problem for quicker convergence / calculation of linear coefficient parameters.  in the Stanford video series, Andrew Ng provides a intuitive explanation and enables one to hone an intuitive understanding.

Multivariate input regression gradient descent converges faster when the inputs are in the same order of magnitude.  For example, if predicting house prices based on X1= the number of rooms and X2= area of the home in square feet.  X1 is on scale of 0-6 bedrooms and and X2 is typically 1000-3000 square feet.  Given the diffence in magnitude, this problem is a good candidate for feature scaling.